This invention relates to modulation schemes for wireless communication. More particularly, the invention relates to the construction of signal constellations for use in unitary space-time modulation of wireless signals.
When wireless communication signals are in transit between a transmit antenna and a receive antenna, they are generally subject to destructive interference and other physical effects that vary in time. As a consequence, the received signal arrives with an attenuation and phase delay that will also generally vary in time. This effect is referred to as fading. Where, e.g., attention is confined to a sufficiently narrow bandwidth, the attenuation and phase delay of the received signal can be described by a complex-valued coefficient h often referred to as the fading coefficient.
Practitioners in the field of wireless communications have recognized that by using multiple antennas for transmission, reception, or both, it is possible to mitigate some of the undesirable consequences of fading, and to achieve certain other benefits as well. For example, the use of multiple antennas affords alternate transmission paths, some of which may, at a given time, be less subject to fading than others. The use of multiple antennas also provides a mechanism for sending redundant signals, the better to understand them at the receiving end, even in the face of severe fading. Even if redundancy is not a primary objective, the use of multiple antennas can provide a mechanism for increasing total transmission rates in a given frequency channel, by simultaneously transmitting multiple, independent signals that can be separated at the receiving end.
FIG. 1 is a simplified, schematic diagram of a wireless communication system having two transmit antennas 10, 15, and three receive antennas 20, 25, 30. As indicated at block 40, baseband-level signals generated at block 35 are modulated onto a carrier wave, which is shown symbolically in the figure as generated at oscillator 50.
It will be seen from the figure that each of the receive antennas receives transmission from, e.g., antenna 10. Provided there is sufficient spatial separation, exemplarily spatial separation, between the receive antennas, the transmitted signals will bear distinct fading effects when they are received. (In this regard, it should be noted that diversity of fading effects can in at least some cases also be achieved by using receive antennas that are selectively receptive to diverse polarizations of the incoming signal, even if the antennas are not substantially separated in space.) A separate fading coefficient hmn accounts for the fading effects, in the physical propagation channel, between each transmit antenna m and each of the receive antennas n. As shown in the figure, m=1 for antenna 10 and m=2 for antenna 15. Similarly, n=1 for antenna 20, n=2 for antenna 25, and n=3 for antenna 30. All six of the fading coefficients hmn are arranged in a matrix H, denoted by block 55 of the figure.
It will be clear from the foregoing that each receive antenna receives, during a given time interval, a total signal that is a weighted sum of the transmissions from the respective transmit antennas. The weight coefficients of that sum are the fading coefficients. The received signal is also typically corrupted by additive noise, which is not indicated in the figure. Because each of the receive antennas is typically receiving a different weighted sum of the transmitted signal, it is theoretically possible under certain conditions to recover the transmitted baseband-level signals by taking appropriate weighted combinations of the demodulated, received signals. One condition necessary for such recovery is that there must be at least as many receive antennas as there are transmit antennas. Another such condition is that the additive noise must not be excessive relative to the signal strength. (It should be noted in this regard that practical methods of signal recovery often imply indirect methods such as maximum-likelihood detection, which is described below.)
Turning again to FIG. 1, demodulation of the received signals is indicated at blocks 60-70. Signal recovery is indicated at block 75. The recovered signals are indicated at block 80.
A new kind of transmitted signal, referred to as a space-time signal has been shown to offer potential improvements in both fading performance and transmission rate. In space-time modulation, each signal that is sent is selected from a finite set, or constellation, of predetermined signal matrices. Thus, if there are a total of L such matrices, each individual matrix that is transmitted conveys log2 L bits of information. Advantageously, the signal matrices are unitary. In a unitary matrix, the columns or rows (whichever are longer) are mutually orthogonal and have unit norm. The individual elements of a unitary matrix are complex numbers; i.e., numbers that are real, imaginary, or sums of real and imaginary components. When unitary matrices are used, the modulation method is referred to as unitary space-time modulation (USTM).
The method of transmitting space-time matrices will now be explained with reference to FIG. 1. A 2xc3x972 space-time signal matrix is represented in block 35 as the matrix:       (                                        s            11                                                s            12                                                            s            21                                                s            22                                )    .
Each element of this matrix is a complex number. Such an element is exemplarily modulated onto the carrier wave by subjecting the carrier wave to a suitable pulse-shaping function of the corresponding complex amplitude and having a width appropriate to the length of a transmission time interval. Each such transmission time interval is referred to as a channel use. During the first channel use, element s11 is transmitted by antenna 10 and element s12 is transmitted by antenna 15. During the second channel use, element s21 is transmitted by antenna 10, and element s22 is transmitted by antenna 15. More generally, each row of a space-time matrix corresponds to a respective channel use, and each column corresponds to a respective transmit antenna. Thus, the entry in row p and column q is the complex amplitude transmitted during the p""th channel use by the q""th antenna. The length of a channel use is generally chosen to be no longer than a fading interval; i.e., a length of time over which the fading coefficients can be assumed constant.
At the receiving end, the transmitted signal matrix is recovered as the matrix       (                                                      s              ^                        11                                                              s              ^                        12                                                                          s              ^                        21                                                              s              ^                        22                                )    ,
as indicated at block 80 of FIG. 1. There are various methods for recovering an estimate of the transmitted signal matrix, based on the received matrix. According to one such method, referred to as maximum likelihood (ML) detection, a likelihood score is computed for each candidate signal matrix that might have been transmitted, and that candidate which maximizes the likelihood score is identified as the transmitted matrix. Typically, the likelihood score is the probability of the raw, basebanded signal amplitudes that are received, given that the candidate matrix was transmitted.
When each signal matrix is transmitted as a matrix drawn from the signal constellation, the ML detector must typically be provided with values of the fading coefficients. Such values may, for example, be measured using appropriate pilot signals. However, there are alternative transmission techniques, referred to as unknown channel techniques, that do not require the ML detector to know the (approximate) fading coefficients, provided the fading coefficients do not change substantially over at least as many, typically at least twice as many, channel uses as there are transmit antennas. One class of unknown channel techniques is referred to as differential modulation. An example of differential modulation is described in greater detail below. Briefly, each signal to be transmitted is the product of the previously transmitted signal, times a new signal matrix selected from the signal constellation. In that case, a suitably adapted ML detector can take advantage of the commonality of fading effects between each signal and its predecessor to effectuate signal recovery without explicit knowledge of the fading coefficients.
One relatively early study of space-time signals based on trellis codes, for transmission over a known channel, is described in V. Tarokh, et al., xe2x80x9cSpace-time codes for high data rate wireless communication: Performance criterion and code construction,xe2x80x9d IEEE Trans. Info. Theory 44 (1998) 744-765.
Unitary space-time modulation, applicable to both known and unknown channels, is described, e.g., in the co-pending U.S. patent application Ser. No. 09/134,297, filed on Aug. 14, 1998 by B. M. Hochwald et al. under the title, xe2x80x9cWireless Transmission Method for Antenna Arrays, Having Improved Resistance to Fading,xe2x80x9d the co-pending U.S. patent application Ser. No. 09/206843, filed on Dec. 7, 1998 by B. Hochwald et al. under the title, xe2x80x9cWireless Transmission Method for Antenna Arrays Using Unitary Space-Time Signals,xe2x80x9d and the co-pending U.S. patent application Ser. No. 09/528973, filed on Mar. 21, 2000 by B. Hassibi et al. under the title, xe2x80x9cMethod of Wireless Communication Using Structured Unitary Space-Time Signal Constellations,xe2x80x9d all commonly assigned herewith.
To help make unknown-channel multiple-antenna communication practical, a differential multiple-antenna modulation scheme using differential unitary space-time signals has been proposed that is well-tailored for unknown continuously varying Rayleigh flat-fading channels. Differential unitary space-time signals are unitary matrix-valued signals that are a multiple-antenna generalization of the standard differential phase-shift keying (DPSK) signals commonly used with a single antenna over an unknown channel. Differential unitary space-time modulation schemes are described, e.g., in the co-pending U.S. patent application Ser. No. 09/356387, filed on Jul. 16, 1999 by B. Hochwald et al. under the title, xe2x80x9cMethod for Wireless Differential Communication Using Multiple Transmitter Antennas,xe2x80x9d commonly assigned herewith.
There remains a need for a principled approach to designing space-time signal constellations that will more fully achieve their theoretical benefits of high data rate and low error probability.
We have found such an approach. According to our new approach, signal constellations of unitary space-time matrices are constructed to have group properties under the operation of matrix multiplication. A set G is a group under a binary multiplication operation if it is closed under multiplication, satisfies the associative law, has an identity element, and contains a multiplicative inverse for each element.
As noted above, the matrix that is detected at the receiving end of the communication system bears the effects of fading and of additive noise. As a consequence, there is some likelihood of error, at the receiver, in ascribing a given received signal to a particular transmitted signal. One important fcature of a space-time signal constellation is that each pair of signal matrices should differ so distinctly from each other that they are unlikely to be confused by the receiver even in the presence of fading and noise.
We have found that signal constellations derived from certain types of groups have a relatively high diversity product. The diversity product is a measure, discussed in more detail below, of how distinguishable the signals are after transmission over a fading channel. Thus, such signal constellations are particularly advantageous for use in space-time modulation.
In one broad aspect, our invention involves a method for wireless signal transmission of signals, in which each signal to be transmitted is selected from a constellation of unitary space-time signals. In some embodiments of the invention, the signal constellation forms a group. In other embodiments, the signal constellation does not contain every member of a group, but a group, referred to as a multiplicative closure, is formed by the set of all possible products of the members of the signal constellation. In still other embodiments, the signal constellation is derived from a group or from a subset of the group by multiplying each member of the group or group subset by a common element, which will typically be a unitary matrix. The resulting constellation is said to be a coset of the group or group subset. In still other embodiments, the signal constellation is an extension of any of the preceding types of constellations, formed by adding one or more further elements that do not belong to and are not derived from the group or group subset.
Significantly, the group in each of these cases is a non-Abelian group of matrices having a positive diversity product. A group is non-Abelian if it contains at least two elements A, B, for which ABxe2x89xa0BA.
In another aspect, our invention involves signal constellations that relate to sets which, although not groups themselves, are constructed according to extensions and generalizations of our rules for constructing group-based constellations.